iorfus said:
Okay, I see ... I had thought about it too (adding a new point afterward) and it did not help me.
You are right, there is a lot of stuff out there. I read some papers and took a look at many\ others. None of them helped me, because I just need
to prove that
1-The solution exists (I did it appealing to Weierstrass theorem)
2-A solution is a point on which the resultant force is null (I just observe that the gradient is null, and the gradient is the resultant force in this case)
And some other things to prove, that I managed to do.
I have seen all the sophisticated algorthms to actually calculate the point, proof that they converge etc.
Interesting, of course. but for my assigment to be complete I only need to extend the proof of point 2 to the case in which the minimum coincides with one of the point. I find no hint on the web ...
If you, Ray, come up with any idea to help me compete this homework, I will be glad to know ... similarly, if anyone has any hint, please let me know!
In the meantime, thanks Ray!
At a non-degenerate solution, the gradient exists and equals the 0 vector. At a degenerate point there is no gradient, so you cannot say that the gradient = 0 there. However, there is a so-called subgradient, and a subdifferential (which is a set of subgradients), and the optimality condition is that the zero vector is a member of the subdifferential set. Alternatively, the function is increasing along any directional derivative at the solution point.
For existence and uniqueness, it is enough to know that the function is strictly convex and that the minimum value inside some large circle of radius ##r## around the origin is smaller than any value outside the circle (so that if there is a minimum at all, it is inside the circle). Existence of a minimum inside the circle follows form Weierstrass, and uniqueness follows from strict convexity.
As for the "forces" business, I don't see the point, but if you were told to do it I guess you have no choice.
The issue is really what happens if the solution is degenerate, say at point ##\vec{y_n}##. In a Varignon frame (where forces are real and are responsible for the solution), the knot comes to rest on the top of the hole at ##\vec{y_n}##; it is too large to be sucked down through the hole, so it stops there. If you force the knot to move away from ##\vec{y_n}## a distance ##\Delta r## along a horizontal unit vector ##\vec{p}##, that raises the weight ##w_n## below ##\vec{y_n}## by ##\Delta r## and needs work input ##w_n \Delta r##. At the same time, some other distances are increased or decreased a bit, so some other weights are lowered or raised a bit. That means that some of the other work inputs may be positive and some may be negative. However, the net work input must be > 0 because the objective is minimized at ##\vec{x} = \vec{y_n}##. You could have a positive work input for
any direction ##\vec{p}##, so the forces would not be balanced.
The forces acting on the knot are (i) those due to the other (n-1) strings, which act horizontally; (ii) the force ##w_n## acting straight down under ##\vec{y_n}##; and (iii) the reaction force of the hole's rim, which prevents the knot from slipping down the hole. The total of all three types of forces must = 0, because the knot does not move; however, there is no reason to assume that the sum of forces (i) and (ii) alonegive zero. (Note that the reaction force must have a horizontal component to cancel the horizontal total of the first (n-1) forces; and, of course, its vertical component is ##+w_n## to cancel the force ##-w_n## due to the hanging weight.)